GARCH (1,)

[narayananvenkat]

david...

In the following GARCH eqns

1. σ2 n-1= .83 + .05µ2 n-1+.93σ2 n-1
2. σ2 n-1= .06 + .04µ2 n-1+.95σ2 n-1
3. σ2 n-1= .60 + .10µ2 n-1+.94σ2 n-1
4. σ2 n-1= .03 + .03µ2 n-1+.93σ2 n-1

locate among the following choices the CORRECT statement:

I. Equation 1 is a stationery model
II..Equation 2 shows nom mean reversion
III Volatility will revert to long run mean level faster than with Eqn 1, than it will for Eqn 4
IV Volatility will revert to a long run mean level faster with Eqn 3 than it will for Eqn 2

VENKAT

 

[David Harper CFA FRM]

Hi venkat,

(convariance) stationary (with an 'a') requires alpha + beta < 1.0. What Hull calls "stable" (but this is outside FRM scope). Since for Equation (1), a+b = .93+.05 = .98, (1) is (covariance) stationary.

Equation (2) does show mean reversion: the first term .06 (omega) = long run variance * weight, so there is mean reversion here

Regarding Eq (4), we can find the *weight* applied to the LR variance with gamma = 1- alpha - beta, since the weights must sum to 1.0. So, Eq (4) gamma = 1 - .93 - .03 = 0.04. As Eq (1) gamma = 0.02, Eq (4) has more rapid mean reversion

And Eq (3), since a+b > 0 is unstable ... so only Eq (1) is true

David